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Appendix I. Subharmonic and Plurisubharmonic Functions

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Appendix I. Subharmonic and Plurisubharmonic Functions Appendix I. Subharmonic and Plurisubharmonic Functions Subharmonic functions and potential theory are often used in the theory of one complex variable. For holomorphic functions of several complex vari­ ables defined in a domain Q c <en, this same important role is played by another class of real valued functions, the class PSH (Q) of functions pluri­ subharmonic in Q. For f E.Yf(Q), both If I and log If I belong to PSH(Q); for CPiEPSH(Q), i=l, ... ,N, sup CPi is in PSH(Q). Thus the class PSH(Q) is a 1 -:5,i5:N natural extension of the set {log If I, f E.Yf(Q)} and gives general methods for the study of this set. Definition 1.1. Let QcJRm be a domain. A real valued function cp(x) with values in [ - 00, + (0) is said to be subharmonic in Q if i) cp(x) is upper semi-continuous and cp(x) $ - 00; ii) for every XEQ and every r<dQ(x) = inf {llx -x'II: X' EC Q} cp(X):;:;W,;;l S cp(x+rrx)dwm(rx)=)o(x. r, cp) lIall ~ 1 where dW m is the Lebesgue measure on the unit sphere sm-l and Wm is the total mass of sm-l. We denote by SeQ) the family of subharmonic functions in Q. If cp and - cp are subharmonic in Q, we say that cp is harmonic in Q. Remark. If cp is subharmonic in Q and l' < dQ(x), then cp(X):;:;,,;;l r- 2m S cp(x+xl)d'm(x/)=A(x, r, cp) Ilxll ;:;;r where dIm is the Lebesgue measure in JRm and 'm is the total mass of the unit ball B(O, 1). Definition 1.2. Let Q c <en be a domain. A real valued function cp(x) with values in [- 00, + (0) is said to be plurisubharmonic if i) cp(z) is upper semi-continuous and cp(z) $ - 00; ii) for every l' such that {z+uw: lui :;:;1', UE<e} cQ 2" cp(z);;;;(2n)-1 S cp(z+rei8 w)dO. o Appendix I. Subharmonic and Plurisubharmonic Functions 231 If q> and - q> are plurisubharmonic, we say that q> is pluriharmonic in Q. We denote by PSH(Q) the family of functions plurisubharmonic in Q. Remark 1. If Qc(Cn, then PSH(Q)cS(Q) and if n=1, PSH(Q)=S(Q). Remark 2. q> c PSH (Q) if and only if it is upper semi-continuous, not identi­ cally - 00, and if its restriction to every complex line Ll meeting Q is either subharmonic or the constant - 00 on each connected If-open set of Ll "Q. Examples. 1) Any continuous convex function in Q (in terms of the underly­ ing real variables) is in PSH(Q), for then we have q>(x) ~H<p(x + y) + q>(x - y)], and if we replace' y by yeiB and integrate with respect to dG we obtain ii) of D e flllltlOn· .. 12. ; 2n 2) if fE:Yf(Q), then 10glfIEPSH(Q); it is enough to show that <p(u)= 1 2rr . f(z+uw) satisfies 10glq>(0)1~- S q>(re'O)dG when the disc {z+uw: lul~r} .. n 2n 0 IS In ••. k If q>(u) == 0, then the inequality is trivial. If not, we let <p(u) = n (u -a) g(u) j=l for lul~r, where g(u) is holomorphic and has no zeros for lul~r. Then log Ig(u)1 is a harmonic function, and since 211: (2n)-1 S 10glreiO-ajldG=sup(loglajl,10gr)~10gla) o we have 2rr k (2n)-1 S 10glq>(reiB)ldG~ I 10glajl+loglg(0)1=10glq>(0)1. o j=l Proposition 1.3. i) If q>E PSH(Q) and c > 0, then Cq>E PSH(Q). ii) If q>l and q>2 are in PSH(Q), then sup (q>l' q>2)EPSH(Q). iii) rr q>v is a decreasing sequence of plurisubharmonic functions in Q, then either lim q>v(z)== - 00 or q>(Z) = lim q>v(z)EPSH(Q). Proof These are immediate consequences of Definition 1.2. o Definition 1.4. A function q>ES(Q) will be said to be continuous if it is continuous for the completion of the Euclidean topology on' IR to the point -00. Remark. q>ES(Q) is continuous if and only if exp q>(x) is continuous for the Euclidean topology on IR. PropositionI.5. If QcIR»! and q>E~2(Q), then q>ES(Q) if and only if m ij2 L1 q>(x)~O, where L1 is the Laplacian L1 = i~l ox;- If Qc (Cn and q>E~2(Q) then 232 Appendix 1. Subharmonic and Plnrisubharmonic Functions <pEPSH(Q) if and only if Proof We write the Taylor series expansion of <p(x) Then since A(x,r,xj-x)=O for all j by symmetry (this is an odd function) and ).(x,r,(xj-x)(X~-Xk))=O forjtok, again by symmetry. Th~ 1 lim [A(X, r, <p) - <p(x)] r- 2 = ~ LI <p(x);:;; o. r~O 2m On the other hand, if Will is the measure of the unit sphere sm-I in ]Rill, w",=2n"'i 2 [F(m/2)]--I, then we obtain from Gauss' Theorem (or Green's Theorem) ,. (I,1 ) A(x,r,<p)=<p(x)+St-",+ldt S Ll<p(x')dT",(x'). o B(x.t) Thus Ll <p(x);:;;O implies that Jc(x, r, <p);:;;<p(x). It follows from Remark 2 that <pE PSH(Q) n yj2(Q) is plurisubharmonic if and only if (1,2) for every WE<C". D Proposition 1.6. If Qe]Rm and <pES(Q)nyj2(Q), then J.(x,r, <p) and A(x, r, <p) 2 are increasing in r and convex functions of um(r)= _r - m for m>2, of u2 (r) =logr if m=2. If Qe<C" and <pEPSH(Q)nyj2(Q), then ).(z, r, <p) and A (z, 1', <p) are increasing with l' and convex in log r. Proof It follows from (I,1) that 8), 8},(x, 1', <p) (m - 2)rm - 1 -;;- (x, 1', <p) or 8u m (r) is increasing, which shows the first part for m > 2. For m = 2, we have 8), 8).(x,r,<p) 1'-8 (x,r,<p) . r c!u z (1') Appendix 1. Subharmonic and Plurisubharmonic Functions 233 If Qden and qJEPSH(Q)n~2(Q), then it follows from (1,1) for m=2 and (1,2) that a 2" . v(z, z', r)=-a-l- S qJ(z+re'Oz')dO ogr 0 is increasing in r. Since this is true for all z', v(z, r) = W Zn1 S v(z, z', r)dw2n (z') =~ Jc(z, r, qJ) Ilz'II=1" ologl is an increasing function of log r, hence },(z, r, qJ) is convex in log r. 0 Remark. The result that Jc(z, r, qJ) is increasing and convex in logr is an important property of plurisubharmonic functions and is not in general true for IR. 2n-subharmonic functions. In the definition of subharmonic and plurisubharmonic functions, we require only upper semi-continuity, whereas in Propositions 1.5 and 1.6, we made assumptions on the regularity of the functions. We now extend these properties in a way to drop the regularity assumptions. Lemma 1.7. Let Q c IR. m be a domain and 0 < c ~ 1. Suppose that Q' c Q and for XEQ', B(x,cdQ(x))cQ'. Then either Q'=Q or Q'=0. Proof By the hypothesis, Q' is open. Suppose that xoEQ' n Q and let d=dQ(xo)' Then there exists a point X'EQ' such that Ilx' -xoll ~cd/4. This implies that dQ(x')~3d/4 and xoEB(x', cdQ(x'))cQ'. Thus Q' is also closed, and since Q is connected, Q' = Q or Q' = 0. 0 In <c n, we denote by D z. w the disc Dz,w= {Z'E<c n : z' =z+uw, UE<C, lui ~ I}. Lemma 1.8. Let Qc<cn be a domain. For zEQ, we set S(z, Q)= U Dz,w' Dz,w cQ Then S(z, Q) is open and if Q' c Q has the property that ZEQ' implies that S(z,Q)cQ', then Q'=Q or Q'=0. Proof Obviously, S(z, Q) is a disked neighborhood of z. Let ZoES(Z, Q) be such that Zo =l= z. Then Zo = z + z' with z' =l= 0 and the disc. D z •z ' is compact in Q. There exists a disked open neighborhood U of the origin such that Dzoz' +UcQ. But Dz,z'+U is a union of discs centered at z, so Dz,z' + U c S(z, Q). Thus S(z, Q) contains an open neighborhood of Zo and hence IS open. To prove the second part of the lemma, we note that S(z, Q) contains the ball B(z, dQ(z)) and so the conclusion follows from Lemma 1.7. 0 Proposition 1.9. If Qc<cn=IR.2n is a domain, then PSH(Q)cS(Q)cL\ocCQ). 234 Appendix I. Subharmonic and Plurisubharmonic Functions Proof Let N be the set of points in Q such that S cpdT Zn = - 00 for every u neighborhood U of z,U~Q. For zEN and [[z'-z[[<idQ(z), the ball B' = B(z', ~ dQ(z)) is compact in Q and is a neighborhood of z.
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